The integrating factor of the differential equation $x \cdot \frac{dy}{dx} + 2y = x^2$ is $(x \neq 0)$.

Let $x=x(y)$ be the solution of the differential equation $2 y e^{x / y^{2}} d x+\left(y^{2}-4 x e^{x / y^{2}}\right) d y=0$ such that $x(1)=0$. Then,$x(e)$ is equal to

Let the solution curve $x=x(y), 0 < y < \frac{\pi}{2}$,of the differential equation $(\log_e(\cos y))^2 \cos y dx - (1+3x \log_e(\cos y)) \sin y dy = 0$ satisfy $x(\frac{\pi}{3}) = \frac{1}{2 \log_e 2}$. If $x(\frac{\pi}{6}) = \frac{1}{\log_e m - \log_e n}$,where $m$ and $n$ are co-prime,then $mn$ is equal to $.....$.

$A$ function $y = f(x)$ satisfies $(x + 1)f'(x) - 2(x^2 + x)f(x) = \frac{e^{x^2}}{(x + 1)}$. If $f(0) = 5$,then $f(x)$ is:

Let $y=y(x)$ be the solution of the differential equation $(x^2+4)^2 dy + (2x^3y+8xy-2) dx = 0$. If $y(0)=0$,then $y(2)$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 3(\tan^2 x + 1)y = \sec^2 x$,with the initial condition $y(0) = \frac{1}{3} + e^3$. Then $y\left(\frac{\pi}{4}\right)$ is equal to: